Physics Experiment Help (Torque, Angular Momentum, etc)

[OrangePotatoCat]

< Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown >

Hey,

So I am not sure if this is in the right section but feel free to move it.

We are conducting an experiment at school at the moment and are having difficulty understanding all the theory and principles as to why these things are happening. However the greatest difficulty we are having so far is determining a way to calculate the theoretical torque for our experiment. Ill post a photo below to help but I was just wondering if there is a way to calculate the torque required for the lever with a force sensor on the left hand side to pull the spinning disc that is attached to a dc motor with varying voltages, out of its axis.
http://imgur.com/TpjqGaK
http://imgur.com/TpjqGaK
The variables we have identified that we know are the diameter of the disc, its weight and the rpm that it is spinning at, we also know the length of rod connecting the motor and disc to the horizontal bar at the top, the length of the horizontal bar and the length of the lever we are pulling.

Also the disc is spun by a motor and then the wooden block that houses the motor and spin left and right 360 degrees which then the horizontal rod can move to act like a pendulum movement.

So the main thing we are struggling with is determining an equation of some sort we can use to get an approximate theoretical torque required to pull the disc out of its axis that we can compare to what data we actually get.

Any idea, thought or improvement is greatly appreciated

 

[paisiello2]

Maybe you could label your picture? I could not follow everything you are describing although I'll guess you are trying to measure some kind of gyroscopic effect:

1) not clear where the pendulum moment comes from and which direction is left and right
2) not sure what you mean by "pull out of its axis.

 

[OrangePotatoCat]

https://i.imgur.com/8maHUhN.jpg

I tried to label the diagram, as best I could with different colours representing a different planes of rotation.
With green being the "pendulum", yellow being the rotation of the disc and red being the rotation of the block.

Then with the blue arrow that is what we are measuring, by keeping that disc perpendicular to the desk and then pulling on the rod with the force sensor attached.
That is where we are trying to determine if there is a way to calculate that theoretical torque and compare it to what we actually get.

 

[paisiello2]

Thanks for the mark up, it helps define things. The only thing I don't see is the rod with force sensor. Can you show that on the picture as well?

 

[haruspex]

Am I right to assume that in any given run of the system the block (red arrow) is in a fixed orientation?
(If not, it's going to get dreadfully complicated, as the red and blue rotations come into and out of phase.)
What equations can you quote regarding torques and angular speeds in gyroscopes?

 

[OrangePotatoCat]

Thanks for the mark up, it helps define things. The only thing I don't see is the rod with force sensor. Can you show that on the picture as well?

The force sensor is attached to the rod, the blue arrow starts from the force sensor

Am I right to assume that in any given run of the system the block (red arrow) is in a fixed orientation?
(If not, it's going to get dreadfully complicated, as the red and blue rotations come into and out of phase.)
What equations can you quote regarding torques and angular speeds in gyroscopes?

Just to clear it up the red arrow does not begin to rotate until a force is applied to the blue arrow by pulling it towards us (direction of the arrow) and then the blue arrow and green arrow move together in a parallel planeAlso if it makes it easier to determine an equation we can just assume they stay in phase to ease the maths involved.

 

[haruspex]

The force sensor is attached to the rod, the blue arrow starts from the force sensor
Just to clear it up the red arrow does not begin to rotate until a force is applied to the blue arrow by pulling it towards us (direction of the arrow) and then the blue arrow and green arrow move together in a parallel planeAlso if it makes it easier to determine an equation we can just assume they stay in phase to ease the maths involved.

Still not making much sense to me.
Pretend the disc is not rotating. You pull on the blue arrow rod (can't actually see a rod, is it obscured by the blue arrow?).
As soon as the support for the force sensor shifts from vertical, there will be a force necessary to counter the weight of the sensor.
The motor block holding the disc may not be positioned with its mass centre on the 'red' axis, so it will tend to rotate.
Now you add the spinning of the disc on top of these unknowns, and the whole thing becomes a mess.
What am I misinterpreting?

 

[TSny]

Is it something like I've tried to show below?

The spinning disc and motor will have a combined angular momentum vector $\vec{L}$ in the y direction of the axes indicated. A force $\vec{F}$ applied in the y direction at the force sensor will create a torque $\vec{\tau}$ in the x direction. This torque will cause $\vec{L}$ to "precess" such that the box (that the disc is attached to) will begin to rotate about the z axis (clockwise looking down on the box).

Is this basically what is going on?

I do not understand the meaning of "pull the disc out of its axis".

 

[OrangePotatoCat]

Is it something like I've tried to show below?

The spinning disc and motor will have a combined angular momentum vector $\vec{L}$ in the y direction of the axes indicated. A force $\vec{F}$ applied in the y direction at the force sensor will create a torque $\vec{\tau}$ in the x direction. This torque will cause $\vec{L}$ to "precess" such that the box (that the disc is attached to) will begin to rotate about the z axis (clockwise looking down on the box).

Is this basically what is going on?

This is essentially what is going on yes, and we want to get an equation so we can calculate the force that is exerted of the brown F. Or the force required

 

[TSny]

This is essentially what is going on yes, and we want to get an equation so we can calculate the force that is exerted of the brown F. Or the force required

The force required to do what, specifically?

 

[OrangePotatoCat]

The force required to do what, specifically?

Considering we have all the center of mass of everything, the disc when spun should sit in this axis until acted upon by another force. Which is the blue arrow we have drawn. So the force required to shift the disc and block in the green arrows drawn above. If this is how the theory works and is possible.

 

[andrevdh]

This might help?

Vector quantities: the net torque applied at the force sensor (blue arrow) - τ_net - and the angular momentum of the spinning disc L end vector quantities
are related by

τ_net = dL/dt

that is the applied torque to the spinning disc is equal to the rate of change of its angular momentum

See for instance HyperPhysics website http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html → Precession

In my diagram the force sensor (your blue arrow) tends to rotate the disc anticlockwise upwards with a force F_fs . This sets up a torque τ_out on the disc, which points out of the page. Causing the disc to rotate upwards (green arrow).

 

[TSny]

Considering we have all the center of mass of everything, the disc when spun should sit in this axis until acted upon by another force. Which is the blue arrow we have drawn. So the force required to shift the disc and block in the green arrows drawn above. If this is how the theory works and is possible.

Here's my guess as to what will basically happen when the force is applied at the force sensor. I might be overlooking something.

Initially, there will not be much movement of the box/disc in your green direction (pendulum direction) and the box/disc will begin rotating as shown by your red arrow. The system is sort of "resisting" moving in the green direction due to gyroscopic effect. As the box/disc rotates toward 90 degree in the red direction from its original position, you will get more motion in the green direction.

If you were to start the box/disc already rotated 90 degrees (in red direction) from what is shown in your picture, then the angular momentum of the system would initially be in the x direction of my picture. Then when you apply the force F in the y direction, the system will easily move in the green direction and there will be little, if any, rotation in the red direction.

It seems to me that there is no definite answer to the question "How much force is required to get the system to move in the green direction?". In principle, if the system had no friction anywhere, then any amount of force (no matter how small) would cause the box/disc to begin rotating in the red direction followed by some motion in the green direction. The motion in the green direction would be small if F is small due to gravity trying to swing the pendulum back to vertical.

 

[andrevdh]

As I understand the system it is forced to rotate in the green direction by the lever system, which starts the precession.

 

[TSny]

As I understand the system it is forced to rotate in the green direction by the lever system, which starts the precession.

If the disc has a large angular momentum, then when the force is applied there will be very little motion initially in the green direction. Essentially all of the initial motion will be precession in the red direction. However, there will begin to be motion in the green direction after the disc has precessed through some angle. Motion in the green direction will be easiest after the disc has precessed 90 degrees.

Here's a video of a gyroscope that shows the effect.


Here, the applied torque is about a vertical axis rather than a horizontal axes but the effect is the same. When the torque is first applied about the vertical axis, there is no (or very little) turning of the gyroscope about the vertical axis. Instead, the gyroscope initially precesses about a horizontal axis.

 

[andrevdh]

The photo is not showing it, but since it is a force sensor (the black box on the left) on the rod I think that the disc and block is pulled/pushed in the green directions while a motor spins the disc at a constant rpm in the yellow direction, but only OPC can clear that up. I hear what you are saying, but since it is wood the moment of inertia would most likely be small and would thus require a large rpm to reach a substantial angular momentum, which is unlikely for the construction, but yes I agree with you.

 

[TSny]

The photo is not showing it, but since it is a force sensor (the black box on the left) on the rod I think that the disc and block is pulled/pushed in the green directions while a motor spins the disc at a constant rpm in the yellow direction, but only OPC can clear that up. I hear what you are saying, but since it is wood the moment of inertia would most likely be small and would thus require a large rpm to reach a substantial angular momentum, which is unlikely for the construction, but yes I agree with you.

Yes. As you say, if the angular momentum of the disc is not very large then there would be some initial swing in the green direction as well as the precession in the red direction. I'm still not very clear on what the OP is asking.

 

[OrangePotatoCat]

The photo is not showing it, but since it is a force sensor (the black box on the left) on the rod I think that the disc and block is pulled/pushed in the green directions while a motor spins the disc at a constant rpm in the yellow direction, but only OPC can clear that up. I hear what you are saying, but since it is wood the moment of inertia would most likely be small and would thus require a large rpm to reach a substantial angular momentum, which is unlikely for the construction, but yes I agree with you.

This is exactly what is happening. And as we found out the block moves in the green arrows direction a certain amount and with the force sensor we were measuring how much force was required to push it back into a perpendicular position with the bench that it originally came from before turning on the motor.

However we are just trying to determine if there is an equation that we can use to back up or prove wrong the resulting forces we are obtaining.

As it is an experiment we are changing the velocity of the disc and the to see how this effects the resulting force and this is where the equation is needed to show how a change in one variable will result in an increased or decreased force.

 

[andrevdh]

What I think he is saying is that the system swings up in the green direction when the motor is started up and he wants to know what force is required to push it back down. The rotational equivalent of F = dp/dt that is tau_net =dL/dt might do it.

 

[OrangePotatoCat]

Hi again we have now obtained results and are unsure as to why we get the results we do.
As the velocity of the disc increases the torque is increased at an exponential rate as define by the excel equation which is where we no longer understand why this is happening.

 

[andrevdh]

So, to be clear the torque is applied to bring the rotating system back down (it is not only the disc that is spinning, but also the motor inside of the block).
So what you are doing in effect is changing the axis of rotation of the system back to its original orientation.
Why would spinning the system faster cause it to raise up higher (I assume that is what is happening?)?

 

[Andrew Mason]

Considering we have all the center of mass of everything, the disc when spun should sit in this axis until acted upon by another force. Which is the blue arrow we have drawn. So the force required to shift the disc and block in the green arrows drawn above. If this is how the theory works and is possible.

It is rather difficult to analyze this configuration using conservation of angular momentum because there are only two degrees of freedom of motion (ie. angular momentum of the apparatus is not conserved when the wheel is rotated by the arm). If you were to allow the system to have 3 degrees of freedom by mounting everything on a platform that was free to rotate about a vertical axis, there should be conservation of angular momentum (assuming friction can be ignored).

In that case, I expect there would be a predictable relationship between the amount of applied torque (as measured by the force sensor and the length of the arm*) and the rotation of the platform (about the vertical axis) and of the motor/wheel (about the horizontal bar).

[*One would have to subtract from the applied torque the opposing torque created by the force of gravity on the motor/wheel].

AM

 

[andrevdh]

As I have it there probably is a bearing in the wooden block so that the system
can rotate (precess?) about a vertical axis (the extension of the arm ) as indicated
by the red arrow.

 

[Andrew Mason]

As I have it there probably is a bearing in the wooden block so that the system
can rotate (precess?) about a vertical axis (the extension of the arm ) as indicated
by the red arrow.

So long as it pivots through the centre of mass of the motor/wheel with negligible friction at any angle, that should work. So, if that is the case, the only difficulty is in determining the actual net torque and angular impulse that is applied to the system.

The problem here is determining the actual angular impulse that one is applying to change the angular momentum of the spinning apparatus. It does not require any torque to maintain a constant angular momentum. But in this configuration it takes a constant torque to keep the arm at an angle other than the vertical. None of that torque affects angular momentum. The other issue is the duration of the torque.

When the force is applied to the arm, it should be applied smoothly and over a fixed time period. Since $\vec{\tau} = \frac{d\vec{L}}{dt}, \Delta L = \int \vec{\tau}dt$. If $\vec{\tau}$ is constant, $\Delta L = \vec{\tau}\Delta t$ = angular impulse. Unless the force is constant and endures for a fixed time period, $\Delta \vec{L}$ will not be proportional to $\vec{\tau}$

Even if you had someway of accurately measuring the applied angular impulse, it is still very difficult to analyse because of the gravitational torque that takes effect when the arm is not vertical. Perhaps a chart of the applied angular impulse with the motor/wheel stopped - as a function of angle could be made. Then do the same chart with measurements taken with the motor/wheel spinning. Subtracting the first from the second should give you the net impulse applied to the motor/wheel to alter its tilt angle.

AM

 

[andrevdh]

I will have to read over your post several times and think about it. A thought came to mind (well several actually)
due to what you said in #22. It seems the system rises up after the motor and wheel is spinning (green arrows), something I found
hard to believe and the OP would not answer me on directly. Is this because the system is limited to only 2 degrees
of freedom and not three since it shorts another axis of rotation perpendicular to the other two?